one show s tha t T = S[S]; fro m thi s descriptio n on e show s firs t tha t T ha s n o te -
stable ideals, then use s this to se e that F ®
T ha s no G-stabl e ideals , and the n use s
this fac t t o conclud e th e desire d isomorphis m wit h F g c C[G].
Theorem 6 shows how the extension E DE G determine s the group G (or rather its
scalar extensio n t o F). A s already noted, Theore m 4 shows how the group G(E/F)
determines th e bas e field F fro m E. W e combin e thes e t o ge t th e fundamenta l
theorem o f differentia l Galoi s theory :
Let E be a Picard-
Vessiot extension of F for L. Then there is a lattice inverting bijective correspondence
{E C K C F\K is a differential subfield {with the same derivation)}
{G G(E/F)\G is a Zariski closed subgroup}
given by
K^G(E/K) and G -+ E
Picard-Vessiot intermediate field extensions correspond to normal subgroups.
remarked above, E i s also a Picard-Vessiot extensio n o f any interme-
diate field K. The n one can deduce from Theorem 5 that G(E/K) i s Zariski closed in
G(E/F), an d Theorem4 implies that EG{E'K) = K. KG i s a Zariski closed subgrou p
of G{E/F), the n G G(E/E
an d E
= E
Le t K denot e this latter field.
In Theore m 6 , the rin g T depend s onl y o n K. an d henc e fro m th e isomorphis m o f
that theore m we conclude that
C[G] equals Fgc C[G(E/E G)l an d thu s tha t
G = G(E/E G). D
In additio n t o providing a convenient pat h t o the Fundamenta l Theorem , Theo -
rem 6 also points the way to an approach to the inverse problem of differential Galoi s
theory. Firs t w e stat e th e invers e problem :
Give n a differential field F an d a linear algebraic group G over
the constant field C o f F, find a Picard-Vessiot extension E o f F wit h G(E/F) = G.
We kno w fro m Theore m 6 tha t i f th e invers e proble m i s solved , the n E i s th e
fraction field of a domain T wit h F' ®F T F ®
C[G] (we denote this latter F[Gj]).
At leas t in the case that G is connected, s o that C[G] is an integral domain , thi s can
be show n t o b e equivalen t t o finding a n appropriat e derivatio n o n F[Gy], namel y
one commuting with G and havin g no proper differentia l ideals . I t o f course suffice s
to find suc h a derivatio n o n F[G F] itself .
We conclude this outline with an historical curiosity. I n 1895, on the occasion o f
birthday o f l'Ecole Normale Superieure , Li e gave an address about Galois ,
in which he talked about the developing Galois theory of differential equations . Wit h
a littl e creative anachronism , on e ca n rea d int o thos e comment s o f nearly a centur y
ago th e mai n poin t o f vie w o f th e presen t work :
"De la par exampl e est nee une theorie generates d'integration pou r le s systemes
d'equations differentielle s don t la solutio n la plu s general e s'exprim e e n functio n
particuliere pa r le s formule s qu i definissen t u n group e fini e t continu ; cett e theori e
a un e analogi e frappant e ave c celle d e Galois. "
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